Tools of Unstable Homotopy Theory

4 James’ torsion bound

A fundamental phenomena that is different between stable and unstable homotopy groups of spheres is that the unstable groups for a fixed group have a uniform torsion bound at each prime. The first result in this direction was due to James [james1957suspension], in the last of a series of three papers on the EHP sequence.

Theorem 4.1. (James) There is a \(2\)-local factorization

By induction, we obtain

Corollary 4.2. There is a factorization

In particular \(4^n\) kills the 2-primary torsion in the homotopy groups of \(S^{2n+1}\). Note also that by the EHP sequence, this provides torsion bounds on the homotopy groups of \(S^{2n}\) as well. At odd primes, the work of Cohen–Moore–Neisendorfer shows that \(p^n\) kills the \(p\)-primary torsion for \(S^{2n+1}\).

The conjectured optimal torsion bound at the prime \(2\) is the order of the restriction of the universal line bundle \(\gamma \) in the diagram below, which is known to be a lower bound.

(-tikz- diagram)

In order to prove Theorem 4.1, we will produce the following diagram

(-tikz- diagram)

The square in the diagram is commutative because there are natural homotopies between \(\Sigma f \circ (-1)\) and \((-1)\circ \Sigma f\) for \(f\) any map between suspensions, because one can do the map \(-1\) in the suspension coordinate.

So it suffices to show that the dashed arrows in the diagram above exist. By the EHP sequence it then suffices to show the proposition below

Proposition 4.3. The composites

\[\Omega ^2 S^{n} \xrightarrow {1-\Omega (-1)} \Omega ^2{S^{n}} \xrightarrow {\Omega H} \Omega ^2 S^{2n-1}\]

and

\[\Omega ^2 S^{2n+1} \xrightarrow {2} \Omega ^2{S^{2n+1}} \xrightarrow {\Omega H} \Omega ^2 S^{4n+1}\]

are null.

Proof. The first statement follows from the commutative square

with the additional observation that \(\Omega H\) is a group homomorphism.

For the second statement, note that we have the following commutative diagram

Where \(h\) is the Hilton–Hopf invariants, and \(f\) is the product of iterated Samelson products. In odd degrees, any iterated Samelson product of the identity of length \(\geq 3\) vanishes, so the composite factors as

\[\Omega \Sigma S^{2n} \xrightarrow {1,1,h_2} \Omega \Sigma S^{2n}\times \Omega \Sigma S^{2n}\times \Omega \Sigma S^{4n} \xrightarrow {\Omega 1\times \Omega -1\times \Omega [1,-1]} \Omega \Sigma S^{2n}\]

It follows that \(0 = 1 + \Omega (-1) + \Omega [1,-1]\circ h\). Looping so that everything is a group homomorphism, composing with \(\Omega H\), and using that \(\Omega H(\Omega (-1)) = \Omega H\), we get \(2\Omega H = - \Omega (H \circ \Omega [1,-1] \circ h)\). We claim that \(H \circ \Omega [1,-1]\) is null. \([1,-1]\) is a suspension since it is \(2\)-torsion, and its Hopf invariant lands in a torsion-free group. Write \([1,-1] = \Sigma v\). Then we have a commutative diagram

But \(\Sigma v\wedge v\) factors through \(\Sigma v\wedge 1 = \Sigma ^{2n}v\), and \(\Sigma ^2v\) is null by the EHP sequence.

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